Twisted conjugacy in $S{L}_n$ and $G{L}_n$ over subrings of ${\bar{\mathbb{F}}}_p(t)$

Abstract

Let $\varphi :G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation ${\sim }_{\varphi }$ on $G$ defined as $x{\sim }_{\varphi }y$ if there exists a $z\in G$ such that $y=zx\varphi (z^{-1})$. The equivalence classes are called $\varphi$-twisted conjugacy classes and the set $G/{\sim }_{\varphi }$ of equivalence classes is denoted $\mathcal{R}(\varphi )$. The cardinality $R(\varphi )$ of $\mathcal{R}(\varphi )$ is called the Reidemeister number of $\varphi$. We write $R(\varphi )=\infty$ when $\mathcal{R}(\varphi )$ is infinite. We say that $G$ has the $R_{\infty }$-{\it property} if $R(\varphi )=\infty$ for every automorphism $\varphi$ of $G$. We show that the groups $G=G{L}_n(R),S{L}_n(R)$ have the $R_{\infty }$-property for all $n\ge 3$ when $F[t]\subset R⊊F(t)$ where $F$ is a subfield of ${\bar{\mathbb{F}}}_p$. When $n\ge 4$, we show that any subgroup $H\subset G{L}_n(R)$ that contains $S{L}_n(R)$ also has the $R_{\infty }$-property.

The main theorems for n>2 case are largely improved and are now much more general. The case n=2 is omitted from this version which will appear elsewhere

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• Twisted conjugacy in classical groups over certain domains of Characteristic $p>0$